Global Reliability Sensitivity Indices Research Articles

An innovative adaptive Kriging-based parallel system reliability method under error stopping criterion for efficiently analyzing the global reliability sensitivity index

An innovative adaptive Kriging-based parallel system reliability method under error stopping criterion for efficiently analyzing the global reliability sensitivity index

Global Reliability Sensitivity of the Structure Under Fuzzy Uncertainty and Its Estimations

Global reliability sensitivity (GRS) index under fuzzy uncertainty is defined as the mean of the absolute difference between unconditional failure credibility and conditional failure credibility, which is important to quantify the effect of fuzzy inputs on failure credibility of the structure. The solution of this index requires nested analysis of conditional failure credibility and fuzzy mean, which introduces a large amount of computation. To efficiently estimate the GRS under fuzzy uncertainty, a method combining fuzzy simulation (FS) with fuzzy first-order and second moment (FS–FFOSM) method as well as a fuzzy sequential optimization and reliability assessment (FSORA) method are proposed. Compared with the direct double-loop FS method, in which both the conditional failure credibility and the fuzzy mean are estimated by FS, the proposed FS–FFOSM uses a more efficient FFOSM method to estimate the conditional failure credibility in the inner loop, which improves the computational efficiency of estimating GRS under fuzzy uncertainty. To further improve the computational efficiency, this paper also transforms the estimation of GRS into the solution of the optimization model under the constraint of failure credibility and establishes the FSORA method to solve the optimization model. The FSORA method uses a series of cycles of deterministic optimization and inverse design point analysis corresponding to the required fuzzy reliability index by GRS to reduce the iterations, and then the computational efficiency is further improved. Several examples illustrate the computational efficiency and accuracy of the proposed methods.

Global reliability sensitivity analysis index and its efficient numerical simulation solution in presence of both random and interval hybrid uncertainty

In the presence of both random and interval hybrid uncertainty (RI-HU), investigating global reliability sensitivity (GRS) can identify the effect of random input on the structural safety globally. To this end, this work establishes the GRS index model and its corresponding efficient solution, and the innovation includes three aspects. Firstly, the GRS of the random input is defined by the average absolute difference between the failure probability upper bound (FP-UB) and the conditional FP-UB on the fixed random input under the RI-HU, and it can reflect the effect of the fixed random input on the structural safety. Secondly, the conditional FP-UB on fixing the random input at the realization is approximated by the conditional FP-UB on fixing the random input in a differential interval. Then the conditional probability theorem can be employed to convert estimating the GRS into the state recognition of the samples, which is the by-product of estimating the FP-UB by the numerical simulation. Finally, by the strategy of surrogating the performance function twice, the meta-importance sampling method is developed to efficiently estimate the GRS in presence of the RI-HU. The rationality of the proposed GRS index model and the efficiency of the developed estimation method are fully verified by the numerical and engineering examples.

An efficient method for estimating time-dependent global reliability sensitivity

Time-dependent global reliability sensitivity can quantify the effect of input variables in their whole distribution ranges on the time-dependent failure probability. To efficiently estimate it to help researchers control the time-dependent failure probability, a novel method is proposed. The proposed method transforms the estimation of unconditional-conditional time-dependent failure probabilities into that of the unconditional-conditional probability density functions (PDFs) of the minimum of time-dependent performance function. Firstly, the minimum of time-dependent performance function is evaluated by adaptive Kriging, and its unconditional-conditional fractional moments are estimated by multiplicative dimensional reduction method (M-DRM). Then, the maximum entropy (MaxEnt) constrained by these fractional moments is used to estimate the unconditional-conditional PDFs, on which the unconditional-conditional time-dependent failure probabilities can be obtained. Finally, the one-dimensional Gaussian quadrature is applied to estimate the time-dependent global reliability sensitivity indices. Due to the high efficiency of adaptive Kriging for estimating the minimum of time-dependent performance function, the avoidance of dimensional curse by M-DRM, and the high efficiency of MaxEnt constrained by fractional moments for estimating PDF, the proposed method can reduce the computational cost dramatically.

Global reliability sensitivity analysis based on state dependent parameter method and efficient sampling techniques

In order to efficiently assess the influence of input variables on the failure of the structural systems, an improved global reliability sensitivity analysis (SA) method is proposed in this paper. The new method is based on the state dependent parameter (SDP) method and the efficient sampling techniques. In the new method, the efficient sampling techniques are first used to generate samples that are more efficient for reliability analysis, and then the SDP method is further employed to estimate the global reliability sensitivity index by using the same set of samples as the reliability analysis. Two efficient sampling methods, e.g., importance sampling (IS) and truncated importance sampling (TIS), are employed in this paper, and the strategies of combining these methods with the SDP method for global reliability SA are discussed. Compared with the existing SDP method, the new method is more efficient for global reliability SA of structural systems. Three examples are used in the paper to demonstrate the efficiency and precision of the new methods.

An efficient method for estimating the parameter global reliability sensitivity analysis by innovative single-loop process and embedded Kriging model

Abstract For evaluating the effect of uncertain distribution parameter on the uncertainty of the failure probability function, the parameter global reliability sensitivity index (PGRSI) is researched and the high efficient estimation algorithm is proposed to approximately estimate the PGRSI. The used definition form of the PGRSI is the expectation of absolute differences between the unconditional expectation of failure probability function and the conditional expectation of failure probability function, which involves a triple-loop evaluation. For efficiently estimating the PGRSI, this paper innovatively inducts the Bayes theorem and the law of total expectation in the successive intervals without overlapping in order to convert the triple-loop evaluation to a single-loop one. The number of actual model evaluations of the single-loop Monte Carlo simulation (MCS) is independent of the dimensionality of uncertain distribution parameters, and the single-loop MCS only needs one set of input-output samples. Besides, the single-loop MCS can be regarded as a classification problem which needs to identify the failure or safety states of samples in the one set. To further enhance the computational efficiency of the proposed single-loop process, the adaptive Kriging (AK) model is embedded in the single-loop MCS, where the Kriging model is used to approximate the actual limit state function to classify the samples. The number of actual model evaluations of the Kriging model nested single-loop process only generates in the process of constructing the AK model. The results of case studies illustrate the accuracy and efficiency of the proposed method.

Structural reliability and reliability sensitivity analysis of extremely rare failure events by combining sampling and surrogate model methods

The aim of this article is to study the reliability analysis, parametric reliability sensitivity analysis and global reliability sensitivity analysis of structures with extremely rare failure events. First, the global reliability sensitivity indices are restudied, and we show that the total effect index can also be interpreted as the effect of randomly copying each individual input variable on the failure surface. Second, a new method, denoted as Active learning Kriging Markov Chain Monte Carlo (AK-MCMC), is developed for adaptively approximating the failure surface with active learning Kriging surrogate model as well as dynamically updated Monte Carlo or Markov chain Monte Carlo populations. Third, the AK-MCMC procedure combined with the quasi-optimal importance sampling procedure is extended for estimating the failure probability and the parametric reliability sensitivity and global reliability sensitivity indices. For estimating the global reliability sensitivity indices, two new importance sampling estimators are derived. The AK-MCMC procedure can be regarded as a combination of the classical Monte Carlo Simulation (AK-MCS) and subset simulation procedures, but it is much more effective when applied to extremely rare failure events. Results of test examples show that the proposed method can accurately and robustly estimate the extremely small failure probability (e.g. 1e–9) as well as the related parametric reliability sensitivity and global reliability sensitivity indices with several dozens of function calls.

An efficient method for estimating global reliability sensitivity indices

Global reliability sensitivity index, defined as the average absolute difference between the original probability density function (PDF) of input and the conditional one on the failure event, can measure the effect of input on the failure probability and provide information for the safety design. However, efficiently estimating the global reliability sensitivity index is still a challenge for engineering structures with rare failure events and implicit complex limit state functions. Thus, a novel method by combining the dynamic Kriging with Monte Carlo simulation (abbreviated as DK-MCS) is proposed for addressing this issue. The novelty of DK-MCS is twofold. One advantage is that the DK model is inserted in MCS process for accurately recognizing failure samples from all MCS samples by the iteratively updated DK model instead of the actual implicit limit state function, thus the computational cost of DK-MCS is significantly less than that of MCS while the precision of DK-MCS is the same as that of MCS. The other is that a new criteria is presented for selecting informative training points to effectively refine the DK model, where the informative training points with the largest contribution to the failure probability are selected by the cross validation method. Therefore, the iteration process for building the DK model can converge more rapidly. Several examples demonstrate the proposed DK-MCS method can greatly save computational cost of estimating the global reliability sensitivity indices and keep the same precision as MCS.

Two Efficient AK-Based Global Reliability Sensitivity Methods by Elaborative Combination of Bayes’ Theorem and the Law of Total Expectation in the Successive Intervals Without Overlapping

The global reliability sensitivity index (GRSI) can measure the effect of model input variable on the failure probability of the structure and provide guidance for the reliability-based design optimization. In this paper, to efficiently estimate the GRSI, an equivalent form of the GRSI is derived by elaborative combination of Bayes’ theorem and the law of total expectation in the successive intervals without overlapping, and it not only makes the total computational cost independent of the dimensionality of model inputs, but also avoids approximating the probability density function approximation used in the existing Bayes’ theorem based global reliability sensitivity analysis. For further improving the efficiency of estimating the GRSI by the equivalent form, two algorithms are presented by nesting the adaptive Kriging (AK) into Monte Carlo simulation (MCS) and importance sampling (IS), respectively, which are abbreviated as AK-MCS and AK-IS. Results of one numerical example and four engineering applications show that the number of model evaluations by the AK-IS is less than $\text{2}\% $ of that by direct IS, and the model evaluation number by AK-MCS is less than $\text{4}\% $ of that by direct MCS under the convergent condition. The results illustrate that the proposed methods for estimating the GRSI are practical for engineering applications.

Reliability sensitivity analysis method for time-dependent problem based on first-passage method

The sensitivity index plays a critical role in the design of product and is used to quantify the impact degree of the uncertainty of the input variable to the uncertainty of the interest output. This paper presents a new local reliability sensitivity method and a global reliability sensitivity analysis method of time-dependent reliability problems. Firstly, according to the Poisson's assumption-based first-passage method, the local reliability sensitivity index is directly obtained by calculating the partial derivative of the failure probability to the distribution parameters of input random variable. Then, the moment-independent global reliability sensitivity index of the time-dependent problems is derived based on the concept of moment-independent. Finally, the efficiency and accuracy of the proposed method are verified with the reference results of Monte Carlo simulation.

An efficient global reliability sensitivity analysis algorithm based on classification of model output and subset simulation

The global reliability sensitivity analysis measures the effect of each model input variable on the failure probability, which is very useful for reliability-based optimization design. The aim of this paper is to propose an alternative method to estimate the global reliability sensitivity indices by one group of model input–output samples. Firstly, Bayes formula is used to convert the original expression of global reliability sensitivity index into an equivalent form where only the unconditional failure probability and the failure-conditional probability density function (PDF) of each model input variable are required. All global reliability sensitivity indices can be simultaneously estimated by this new equivalent form, and the computational cost of the process is independent of the dimensionality of model input variables. Secondly, to improve the efficiency of sampling which aims at calculating the unconditional failure probability and estimating the failure-conditional PDF of every model input simultaneously, subset simulation method is extended to achieve these two aims. In the proposed procedure, subset simulation is used to estimate the unconditional failure probability, and Metropolis-Hastings algorithm is employed to convert the samples in failure domain from the current PDF in subset simulation to the PDF corresponding to the original PDF of model inputs for estimating the failure-conditional PDF of each model input variable. Thirdly, Edgeworth expansion is employed to approximate the failure-conditional PDF of each model input variable. Finally, the global reliability sensitivity index can be easily computed as byproducts using the unconditional failure probability and the failure-conditional PDF of each model input in failure probability analysis, and this process does not need any extra model evaluations after the unconditional failure probability analysis is completed by subset simulation. A headless rivet model, a roof truss structure and a composite cantilever beam structure are analyzed, and the results demonstrate the effectiveness of the proposed method in global reliability sensitivity analysis.

Global Reliability Sensitivity Analysis Based on Maximum Entropy and 2-Layer Polynomial Chaos Expansion.

To optimize contributions of uncertain input variables on the statistical parameter of given model, e.g., reliability, global reliability sensitivity analysis (GRSA) provides an appropriate tool to quantify the effects. However, it may be difficult to calculate global reliability sensitivity indices compared with the traditional global sensitivity indices of model output, because statistical parameters are more difficult to obtain, Monte Carlo simulation (MCS)-related methods seem to be the only ways for GRSA but they are usually computationally demanding. This paper presents a new non-MCS calculation to evaluate global reliability sensitivity indices. This method proposes: (i) a 2-layer polynomial chaos expansion (PCE) framework to solve the global reliability sensitivity indices; and (ii) an efficient method to build a surrogate model of the statistical parameter using the maximum entropy (ME) method with the moments provided by PCE. This method has a dramatically reduced computational cost compared with traditional approaches. Two examples are introduced to demonstrate the efficiency and accuracy of the proposed method. It also suggests that the important ranking of model output and associated failure probability may be different, which could help improve the understanding of the given model in further optimization design.

A modified importance sampling method for structural reliability and its global reliability sensitivity analysis

The importance sampling method is an extensively used numerical simulation method in reliability analysis. In this paper, a modification to the importance sampling method (ISM) is proposed, and the modified ISM divides the sample set of input variables into different subsets based on the contributive weight of the importance sample defined in this paper and the maximum super-sphere denoted by β-sphere in the safe domain defined by the truncated ISM. By this proposed modification, only samples with large contributive weight and locating outside of the β-sphere need to call the limit state function. This amelioration remarkably reduces the number of limit state function evaluations required in the simulation procedure, and it doesn’t sacrifice the precision of the results by controlling the level of relative error. Based on this modified ISM and the space-partition idea in variance-based sensitivity analysis, the global reliability sensitivity indices can be estimated as byproducts, which is especially useful for reliability-based design optimization. This process of estimating the global reliability sensitivity indices only needs the sample points used in reliability analysis and is independent of the dimensionality of input variables. A roof truss structure and a composite cantilever beam structure are analyzed by the modified ISM. The results demonstrate the efficiency, accuracy, and robustness of the proposed method.

Extending first-passage method to reliability sensitivity analysis of motion mechanisms

Identifying the parameters that substantially affect the time-dependent reliability is critical for reliability-based design of motion mechanism. The time-dependent local reliability sensitivity and global reliability sensitivity are the two effective techniques for this type of analysis. This work extends the first-passage method and PHI2 method, which are commonly used for estimating the time-dependent reliability, for efficiently estimating the time-dependent local reliability sensitivity and global reliability sensitivity indices of the motion mechanism. Both the local reliability sensitivity and global reliability sensitivity indices are analytically derived based on the Poisson assumption–based first-passage method and the first-order Taylor’s expansion of the motion error function. Compared with the current envelope function method for estimating the time-dependent local reliability sensitivity and global reliability sensitivity indices, the developed method does not need to estimate the second-order derivatives of motion error function, thus is more applicable. The accuracy and effectiveness of the proposed method are demonstrated by a numerical example and a satellite antenna, the direction of which is controlled by a four-bar function generator mechanism.

Time-variant global reliability sensitivity analysis of structures with both input random variables and stochastic processes

The ubiquitous uncertainties presented in the input factors (e.g., material properties and loads) commonly lead to occasional failure of mechanical systems, and these input factors are generally characterized as random variables or stochastic processes. For identifying the contributions of the uncertainties presented in the input factors to the time-variant reliability, this work develops a time-variant global reliability sensitivity (GRS) analysis technique based on Sobol' indices and Karhunen- Loeve (KL) expansion. The proposed GRS indices are shown to be effective in identifying the individual, interaction and total effects of both the random variables and stochastic processes on the time-variant reliability, and can be especially useful for reliability-based design. Three numerical methods, including the Monte Carlo simulation (MCS), the first order envelope function (FOEF) and the active learning Kriging Monte Carlo simulation (AK-MCS), are introduced for efficiently estimating the proposed GRS indices. A numerical example, a beam structure and a ten-bar structure under time-variant loads are introduced for demonstrating the significance of the time-variant GRS analysis technique and the effectiveness of the numerical methods.

Global reliability sensitivity analysis of motion mechanisms

The kinematic failure of a mechanism is commonly caused by the random input errors such as the errors of component dimensions and motion inputs. For identifying the main source of the failure probability, the global reliability sensitivity analysis is introduced. The method is based on decomposing the variance of the failure domain indicator function into partial variances of increasing orders, which enable attributing the failure probability to each of the random input errors and their interactions. Then the analytical solutions of the global reliability sensitivity indices are derived based on the first-order Taylor series expansion of the motion error function. Compared with the traditional local reliability sensitivity indices, the global reliability sensitivity analysis technique has two main merits. First, it is more suitable for ranking the importance of random input errors and identifying the source of failure probability. Second, it provides not only the individual effect of each random input error on the failure probability but also their interaction and total effects. The engineering significance of the global reliability sensitivity indices as well as the effectiveness of the analytical method for computing the global reliability sensitivity indices are demonstrated with a four-bar sine function generator mechanism and a rack-and-pinion steering linkage.

Time-dependent reliability sensitivity analysis of motion mechanisms

Reliability sensitivity analysis aims at identifying the source of structure/mechanism failure, and quantifying the effects of each random source or their distribution parameters on failure probability or reliability. In this paper, the time-dependent parametric reliability sensitivity (PRS) analysis as well as the global reliability sensitivity (GRS) analysis is introduced for the motion mechanisms. The PRS indices are defined as the partial derivatives of the time-dependent reliability w.r.t. the distribution parameters of each random input variable, and they quantify the effect of the small change of each distribution parameter on the time-dependent reliability. The GRS indices are defined for quantifying the individual, interaction and total contributions of the uncertainty in each random input variable to the time-dependent reliability. The envelope function method combined with the first order approximation of the motion error function is introduced for efficiently estimating the time-dependent PRS and GRS indices. Both the time-dependent PRS and GRS analysis techniques can be especially useful for reliability-based design. This significance of the proposed methods as well as the effectiveness of the envelope function method for estimating the time-dependent PRS and GRS indices are demonstrated with a four-bar mechanism and a car rack-and-pinion steering linkage.

Efficient sampling methods for global reliability sensitivity analysis

An important problem in structure reliability analysis is how to reduce the failure probability. In this work, we introduce a main and total effect indices framework of global reliability sensitivity. By decreasing the uncertainty of input variables with high main effect indices, the most reduction of failure probability can be obtained. By decreasing the uncertainty of the input variables with small total effect indices (close to zero), the failure probability will not be reduced significantly. The efficient sampling methods for evaluating the main and total effect indices are presented. For the problem with large failure probability, a single-loop Monte Carlo simulation (MCS) is derived for computing these sensitivity indices. For the problem with small failure probability, the single-loop sampling methods combined with the importance sampling procedure (IS) and the truncated importance sampling procedure (TIS) respectively are derived for improving the calculation efficiency. Two numerical examples and one engineering example are introduced for demonstrating the efficiency and precision of the calculation methods and illustrating the engineering significance of the global reliability sensitivity indices.